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Let $p:C\rightarrow D$, $i:F\rightarrow D$ be functors of 2-categories, and we form the lax pullback of $p$ along $i$ $$ \bar{p}:C\times_D^{lax} F\rightarrow F$$

Q1: Is it true that if $p$ preserves finite limits, then $\bar{p}$ does also?

If it helps, you may assume that $i$ is fully faithful.

Q2: same question for double categories

NB: By 2-categories I mean (any model for) $(\infty,2)$ categories.

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  • $\begingroup$ If everything is a 1-category, then what is true is that if C and D have limits, and i and p preserve them, then the lax pullback has limits and both projections preserve them. $\endgroup$
    – Dimitri Chikhladze
    Commented Apr 7, 2014 at 16:19
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    $\begingroup$ "2.16 Limits in comma categories" in "Francis Borceux, Handbook of Categorical Algebra 1, Cambridge University Press". In Cat lax pullbacks are comma categories. $\endgroup$
    – Dimitri Chikhladze
    Commented Apr 8, 2014 at 9:59
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    $\begingroup$ It depends on what you mean by "lax pullback". The use of "lax pullback" to mean "comma object" is better avoided, since comma objects are not actually any sort of lax limit. See ncatlab.org/nlab/show/2-limit#lax . $\endgroup$
    – Mike Shulman
    Commented Apr 8, 2014 at 15:54
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    $\begingroup$ @AdamGal, the definition I linked to is for 2-categories (i.e. (2,2)-categories) where there is really only one model. If you wanted an analogous definition for $(\infty,2)$-categories, you'd have to formulate it in an appropriate way for your chosen model. $\endgroup$
    – Mike Shulman
    Commented Apr 9, 2014 at 19:57
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    $\begingroup$ I forget who it was who said that the only thing worse than bad terminology is continually changing terminology. Cartesian square also means pullback. $\endgroup$
    – Mike Shulman
    Commented Apr 16, 2014 at 4:33

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